Models of RDD

Last update: 2000 August 12

1. Purpose of modeling RDD

Increase of residual dark distribution (RDD) has various impacts on the SIS data. For example, it can degrade energy resolution, detection efficiency, and change the line profile (see this page for details). When we analyze the SIS data affected by the RDD, the effects need to be corrected in data, or, if it is difficult to correct, need to be adequately included in the calibration data. In either case, RDD appropriate for the observation are necessary. RDD may be calculated from a set of frame mode data. But the frame mode data were obtained typically only once a month. Thus most of the observations do not have corresponding frame mode data. Therefore, we need an RDD model which can approximate RDD for any observations with reasonable accuracy.

2. Calculation of RDD

RDD is calculated from the frame mode data. Although we have been taking the frame mode data regularly since 1995, few frame mode data were taken in 1993 and 1994. For these years, we also use the corner pixel distribution to calculate RDD. It is known that RDD depends on the CCD temperature. We did not use the data obtained when the CCD temperature was high to calculate RDD.

2.1 RDD map

If we calculate a PH distribution of pixels from the raw frame mode data, what we obtain is close to RDD. However, there are two problems in this simple calculation (except for the offset of the zero point), which are (1) dark frame is not subtracted, (2) contaminated by the X-ray data and particle data. To overcome these problems, we use two or more sets of frame mode data. For example, in the simplest case, contaminated pixels may be removed, if we compare two sets of frame mode data pixel by pixel and take a lower PH. In reality, we adopt a more sophisticated method. Details of the calculation method is found in this page. Results of the calculation is called an RDD map.

There is one significant difference between the RDD map calculated above and the RDD. RDD map is usually calculated from about 3-10 sets of frame mode data. This means that RDD maps are basically average of 3-10 frame data (except for pixels contaminated by X-rays or particles). Thus the read-out noise in RDD maps is much reduced. The difference may be corrected when we calculate the RDD model parameters from the RDD maps.

2.2 Corner Pixel Distribution

When frame mode data are not available, we use corner pixel distribution of the event data. If we take event data of grades 0, 2, 3, 4, corner pixels may be considered to have little X-ray contamination and can be used to calculate the RDD. However, from definition of grades, PH of the corner pixels is limited to below the split threshold (40 ADU). This becomes problematic if the hard tail of RDD exceeds 40 ADU. Another problem in the corner pixel distribution is that some of the pixels may have small but significant contamination from the X-ray event charge. Even small contamination may become important when the RDD effects are small. Therefore, we should be careful to use the corner pixel distribution to calculate RDD.

2.3 Examples of RDD

We show examples of RDD here. RDD strongly depends on the read-out modes, in other words, exposure time. Furthermore, S0 and S1 have slightly different distribution. S1 tends to have larger dark current than S0, and the RDD of S1 has more significant hard tail. Difference of RDD among the sensor is not large, although the difference can be investigated only for the 2 and 4 ccd mode data (exposure of 8 sec and 16 sec, respectively).

3. Models of RDD

Two empirical models of RDD are explained below. One is an exponential tail model introduced by A. Rasmussen (ASCA newsletter No.3, "Effects of Radiation Damage on SIS Performance") and the other is a poisson model. The former was the first model adopted to reproduce RDD, but the approximation become poorer for the recent data. Thus we introduced so-called a poisson model which gives reasonably good approximation for overall data.

RDD model is used in the FTOOLS, "faintdfe". Current version of "faintdfe" (Dec 1995) uses the exponential tail model. Modification to use the poisson model is now in progress.

3.1. Model 1: Exponential Tail Model

RDD results from the dark current in pixels. Therefore, distribution of the dark current might represent RDD under some ideal condition (i.e. when readout noise is absent; see below). However, in reality, we cannot measure the dark current directly. We can measure only the dark current superimposed by the readout noise. Mathematically, this process is represented by the convolution of the dark current distribution and the readout noise. In other words, RDD may be represented by the convolution of the dark current distribution and the readout noise. Readout noise has a simple gaussian distribution. It is easily measured by the PH distribution of the overclocked pixels, and is known to be stable since the initial operation of SIS. Thus, if we can model the dark current distribution, we can get the RDD model simply by convolution with a gaussian.

When the radiation damage on the CCD is little, RDD tends to have a gaussian form with a small hard tail. As seen in the previous figure of RDD, the hard tail may be approximated by a straight line in the semi-logarithmic plot, which corresponds to an exponential function. We found that this approximation works quite well for some of the RDD. Thus we adopt a model consisting of a δ-function and an exponential tail for the dark current distribution. Convolution of this model with the Gaussian read-out noise gives RDD, i.e.
,
where f is the fraction of damaged pixels, Q the scale of exponential tail, q₀ the offset of distribution, and q the PH. Because q₀ is determined by the DFE correction, its value does not have practical importance. The principle of this RDD modeling is schematically explained in this figure. This formula can approximate the RDD quite well when the radiation damage is not very large. Some examples of model fitting to the observed RDD are found here. History of the model parameters is shown in this page. As seen from the figures, when the fraction of pixels which have excess dark current become almost unity, the fit to the data become poor. This motivated us to introduce a revised model.

3.2. Model 2: Poisson Model

• Number of defects in a pixel obeys the Poisson distribution.
• All the defects generate the same amount of dark current on average.
• Some amount of defect exist even at the launch time.

A few words need to be added to the third assumption. Because these initial defects are not originated from the radiation damage, they do not obey Poisson distribution. They should be separately included in the model from the defects produced by the radiation damage. Because the defects due to the radiation damage may be still generated in the pixels with initial defect, the effect of the initial defects need to be included to the convolution function. Thus the convolution should be taken not with Gaussian but with Gaussian plus a hard tail, i.e. the exponential tail model.

Although the Poisson distribution is defined only for integer, we regard it as a continuous distribution. Fluctuation of the dark current is considered to make the distribution effectively continuous. Statistical fluctuation may be the main source of the dark current fluctuation, but the difference of each defect also contribute the fluctuation. For this purpose, we substitute a Gauss function instead of the factorial in the definition of the Poisson distribution. Thus the model is defined as follows:
,
where Γ(x) is the Gamma function, which is reduced to factorial (x-1)! for integer. The principle of this RDD modeling is schematically explained in this figure. The model parameters are (1) average number of defects in a pixel (λ), (2) average dark current per defect (J), (3) offset of the distribution (q₀), (4) standard deviation due to the readout noise and residual fluctuation (σ), and (5) normalization factor (T). Parameters of RDD₁ should be fixed to the values at the launch time except for σ. The normalization factor, T, should be equal to the total number of pixels. The offset is determined by the on-board subtraction of the dark frame. When the number of defects is very small, Poisson distribution may be approximated by a δ-function. This means that the Poisson model reduces to the exponential tail model when the radiation damage is small.

Caution should be paid when we interpret the meaning of the model parameters. When we constructed the poisson model, clear physical meaning was assigned to each parameters. However, this does not mean that the best-fit parameters we obtain by fitting the poisson model to the observed RDD still hold the originally assigned meaning. The poisson model is just one of the simplest model to approximate the RDD and probably does not reproduce the RDD exactly. Thus the best-fit parameters may be largely deviated from their true physical values. For example, as we show later, dark current of a defect is very different between 1 ccd and 4 ccd mode. We do not consider that the dark current due to a defect is really very different between 1 ccd and 4 ccd mode. But it is probably an artifact due to the simplified model of RDD.

Examples of the model fitting to the observed RDD are shown here. We found that, when the radiation damage is moderate (asca time ~ 3-5×10⁷sec), some of the parameters (number of defects and dark current) are strongly coupled and are difficult to determine uniquely. Therefore, we fixed the dark current to a constant value. Note that this does not affect the shape of the model function, but just resolve the coupling of the parameters. Histories of the parameters are shown in this page.

4. RDD Model in the Analysis Software

RDD model is used in some of the analysis software of SIS data. The program to calculate the DFE ("faintdfe") explicitly use the RDD model. Note that "faintdfe" may be invoked from some scripts, such as "ascascreen". SIS response generator ("sisrmg") also uses the RDD model to include the change of the line profile. However, current version of "sisrmg" (ver 1.1, April 1997) does not fully incorporate the effects of RDD.

Back